Every complex in \(\mathcal {X}\)  has a resolution by vector bundles \(\mathcal {O}_V\otimes \Gamma \)  for \(\Gamma \)  a \(G\) -representation, so \(\Phi \)  is essentially surjective. Let \(\Gamma \)  and \(\Gamma ^{\prime }\)  be \(G\) -representations. \(\Phi \)  is fully faithful because
\(\text{Ext}^i_{K^b\left(\textbf {A}\right)}\left(\mathcal {O}_V\otimes \Gamma , \mathcal {O}_V\otimes \Gamma ^{\prime }\right)={\left\lbrace \begin{array}{ll}\left(\text{Sym}\left(V^{\vee }\right)\otimes \Gamma ^{\prime }\otimes \Gamma ^{\vee }\right)^G\text{ for }i=0,\\0, \text{ otherwise},\end{array}\right.}\) 
and the computation for \(\text{Ext}^i_{D^b\left(\mathcal {X}\right)}\left(\mathcal {O}_V\otimes \Gamma , \mathcal {O}_V\otimes \Gamma ^{\prime }\right)\)  gives the same answer, see for example [1]}.
